Reduced Forms: Feasibility, Extremality, Optimality

Abstract

We study independent private values auction settings with heterogeneous bidders, revenue nonlinear in interim winning probabilities, and place no ad hoc restrictions on auctions that a seller can use. We show that feasibility of interim winning probabilities is testable along a unidimensional principal curve, constructed in closed form. We then explicitly characterize the extreme points of the feasible set and the allocations that induce them. Combining these results, we solve for the optimal auction under a natural regularity condition. The optimal auction equalizes bidders' marginal revenues along its principal curve and awards the good to the bidder whose type the curve reaches first. Revenue curvature determines whether and how low types are excluded---at the intensive margin, scaling winning probabilities down, or the extensive margin, cutting them off. Applications include the linear model, endogenous valuations from ex ante investments, and non-expected utility preferences, where previous results were largely limited to symmetric environments and allocations or two bidders.